Rotating a Point around Another Point for 90 Degrees: A Comprehensive Guide

When dealing with points in a coordinate system, one common task is to rotate one point around another. This guide will walk you through the process of rotating point M3,4 around S1,3 by 90 degrees. We'll cover both counterclockwise and clockwise rotations, ensuring you have a thorough understanding of the concept and the steps involved.

Step-by-Step Guide to Rotating Point M34 Around S13 by 90 Degrees

To rotate a point M around another point S by an angle theta; (in this case, 90 degrees), follow these steps:

Step 1: Translate S to the Origin

First, translate the pivot point S to the origin (0,0) by subtracting its coordinates from the coordinates of M.

For M(3,4) and S(1,3):

Mark  M  (x - a, y - b) Mark  M  (3 - 1, 4 - 3)  (2, 1)

Step 2: Rotate the Translated Point around the Origin

Use the rotation transformation formula to rotate the translated point M around the origin by theta; (90 degrees in this case).

The rotation matrix for a point (x, y) around the origin by theta; is:

M'  (x cos  theta - y sin  theta, x sin  theta   y cos  theta)

For a 90-degree rotation, the values of cos90deg; and sin90deg; are 0 and 1, respectively:

(x cos 90° - y sin 90°, x sin 90°   y cos 90°)  (-y, x)

Applying this to point M(2, 1):

M'  (-1, 2)

Step 3: Translate Back to the Original Position of S

Finally, translate the rotated point back to its original position by adding the coordinates of S to the rotated point M'.

For M'(-1, 2) and S(1, 3):

M_{text{rotated}}  (M_x   a, M_y   b) Mark  M_{text{rotated}}  (-1   1, 2   3)  (0, 5)

The coordinates of point M3,4 after rotating 90 degrees around point S1,3 are (0,5).

Rotating in Both Directions: Clockwise and Counterclockwise

In addition to the 90-degree rotation in the counterclockwise direction, it's also important to understand how to rotate in the clockwise direction. The main idea is to translate the pivot point S to the origin, perform the rotation, and then translate both points back to their original positions.

Counterclockwise Rotation by 90 Degrees

To rotate point M around S by 90 degrees in a counterclockwise direction:

Step 1: Translate the pivot point S to the origin.

Use the translation formula T(x, y) (x - a, y - b):

T(x, y)  (x - a, y - b) Mark  T(2, 1)  (2 - 1, 1 - 3)  (1, -2)

Step 2: Rotate the translated point around the origin.

Use the counterclockwise rotation formula R90deg;(x, y) (-y, x):

R90deg;(1, -2)  (-(-2), 1)  (2, 1)

Step 3: Translate back to the original position of S.

Use the translation formula T(x, y) (x a, y b):

T(2, 1)  (2   1, 1   3)  (0, 5)

The coordinates of point M3,4 after rotating 90 degrees in a counterclockwise direction around point S1,3 are (0,5).

Clockwise Rotation by 90 Degrees

To rotate point M around S by 90 degrees in a clockwise direction:

Step 1: Translate the pivot point S to the origin.

Use the translation formula T(x, y) (x - a, y - b):

T(2, 1)  (2 - 1, 1 - 3)  (1, -2)

Step 2: Rotate the translated point around the origin.

Use the clockwise rotation formula R-90deg;(x, y) (-y, -x):

R-90deg;(1, -2)  (-(-2), -(1))  (2, -1)

Step 3: Translate back to the original position of S.

Use the translation formula T(x, y) (x a, y b):

T(2, -1)  (2   1, -1   3)  (3, 2)

The coordinates of point M3,4 after rotating 90 degrees in a clockwise direction around point S1,3 are (3,2).

Conclusion

Rotating a point around another point in a coordinate system is a fundamental concept in geometry and computer graphics. By understanding the steps involved in both counterclockwise and clockwise rotations, you can easily manipulate points in various applications. Experiment with these transformations on your own to solidify your understanding!